Contributions: In this paper, inspired by results from graph signal processing, we introduce the notion of Graph Symmetric Systems (GSSs), which are systems that are symmetric with respect to an underlying graph topology (formalized in §-A). We show that for such systems, the resulting Linear Quadratic (LQ) centralized optimal controller admits an efficient message passing implementation in the form of a novel class of graph filters defined by transfer function filter taps. We subsequently propose and analyze two complementary approaches to computing near-optimal distributed controllers by truncating the centralized optimal controller subject to stability constraints. By leveraging tools from robust System Level Synthesis (SLS) [1]}, [2]}, we show that these truncation algorithms can be solved via convex optimization, and that the resulting distributed controllers enjoy sub-optimality guarantees relative to the centralized optimal controller. These results constitute an important first step towards bridging the complementary, but traditionally disparate, fields of distributed optimal control and graph signal processing.
